Any suggestions for improvement are more than welcome.

Maybe it's just me, but wording seems a bit unclear.

This is supposed to be an enhanced Dual Wielding option right? That's what I understand witht the non-restriction on Light weapons. So you're saying a character can now proprerly use 2 non-light weapons (Javelins, Maces, Quarterstaffs, Spears, Battleaxes, Flails, Longswords, Morningstars, Rapiers, Tridents, War picks, Warhammers and Whips) in combat instead of the usual Light Weapons(Daggers, Clubs, Handaxes, Light Hammers, Sickles, Scimitars and Shortswords), which is admittedly awesome, going around with double Spears Lancer style seems like fun. Issue is the Dual Wielder feat already kinda does that " You can use two-weapon fighting even when the onehanded melee weapons you are wielding aren't light. "

The attack is odd though, what does the "make a second weapon attack" part mean? Is it to be treated like an Extra Attack (can attack one additional time with your weapon, applying both dice and modifiers to the damage) or like the Dual Wielding attack (after attacking with main weapon make an additional attack with off-hand weapon using a bonus action and only appyling dice damage)?

Here's a suggestion:

"When you take the Attack action with a weapon you're holding in one hand, you can use a bonus action to attack with a different weapon that you're holding in the other hand,  adding modifiers as usual." -> It's just the Dual Wielding attack wording, but with extra modifiers on the bonus attack. Leaving it as a bonus action means players still have a tradeoff instead of the Extra attack, meaning they won't be able to use that bonus action to do extra stuff. It also means it changes (atleast a bit) the overlapping there is with the Dual Wielder feat, this one means more damage and a score increase on top of the improved dual wielding, while Dual Wielder gives AC (and the draw/stow speed improvement, but not everyone uses those rules) on top of the dual wielding improvement.

Sorry for the text wall, I kept rambling. Hope it helps and maybe it sparks some ideas for the feat. 

Good luck.

Thanks for the rework in the wording.  I wasn't exactly how clear it would be.  I also didn't take the Dual Wielding feat into account, as I don't normally use feats.

This is pretty much a watered down "Dual Wielder" feat turned into a half feat (adding +1 to an ability score).  You should take a look at Dual Wielder, and maybe make this feat a follow up to it with some other kind of bonus.  Right now, the bonus of Dual Wielder would have me pick that over this every time.  +1 to STR or DEX is nice, but the +1 to ac and the ability to draw or stow two one-handed weapons when you would normally be able to draw or stow only one is better in my opinion.

If Dual Wielder wasn't a feat, I'd definitely like this one, but as it already is I find this one a little lackluster. 

I know.   Honestly, if I could delete this one I could.  I don't usually use feats, so I was unaware of Dual Wielder.

Quote from LordShadowthorn >>

I know.   Honestly, if I could delete this one I could.  I don't usually use feats, so I was unaware of Dual Wielder.

 I mean it's no worry.  I personally would just modify it to make it work with Dual Wielder